论文标题
所有三个玩家游戏的并行重复的多项式界限都带有二进制输入
Polynomial Bounds On Parallel Repetition For All 3-Player Games With Binary Inputs
论文作者
论文摘要
我们证明，每3个玩家（3个磁带）游戏$ \ MATHCAL G $，值少于一个，其查询分布具有支持$ \ Mathcal s = \ {（1,0,0,0），（0,1,0），（0,1,0），（0,0,0,1）\} $ hammming witter the the $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $。 g^{\ otimes n} $衰减在多项式上快到零;也就是说，有一个常数$ c = c（\ mathcal g）> 0 $，使得游戏的值$ \ mathcal g^{\ otimes n} $最多是$ n^{ - c} $。 遵循Girish，Holmgren，Mittal，Raz和Zhan（STOC 2022）的最新工作，我们的结果是缺失的作品，暗示着类似的多人游戏类似的多人游戏：$ \ textbf {everther} $ 3-玩具$ \ ship textiT $ \ textiT $ \ textiT $ \ textiT $ \ by textiT $ \ by textiT $ \ by textiT lturn and $ {$ {$ {比1，有一个常数$ c = c（\ mathcal g）> 0 $，使得游戏的值$ \ mathcal g^{\ otimes n} $最多是$ n^{ - c} $。 我们的证明技术是新的，需要许多新想法。例如，我们利用布尔傅里叶分析的级别$ k $不平等现象，据我们所知，在我们的工作之前，在这种情况下尚未探索。
We prove that for every 3-player (3-prover) game $\mathcal G$ with value less than one, whose query distribution has the support $\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $\mathcal G^{\otimes n}$ decays polynomially fast to zero; that is, there is a constant $c = c(\mathcal G)>0$ such that the value of the game $\mathcal G^{\otimes n}$ is at most $n^{-c}$. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more general class of multiplayer games: For $\textbf{every}$ 3-player game $\mathcal G$ over $\textit{binary questions}$ and $\textit{arbitrary answer lengths}$, with value less than 1, there is a constant $c = c(\mathcal G)>0$ such that the value of the game $\mathcal G^{\otimes n}$ is at most $n^{-c}$. Our proof technique is new and requires many new ideas. For example, we make use of the Level-$k$ inequalities from Boolean Fourier Analysis, which, to the best of our knowledge, have not been explored in this context prior to our work.
